Carnot’s Cycle

Carnot s cycle A hypothetical scheme for an ideal heat machine. Shows that the maximum efficiency for the conversion of heat into work depends only on the two temperatures between which the heat engine works, and not at all on the nature of the substance employed. 

Kuhn, T. S. (1955). Carnot s Version of Carnot s Cycle. American Journal of Physics 23 91-95. 

There is now performed a reversible cycle called Carnot s cycle, and consisting of four operations ... 

We shall now apply the two laws of thermodynamics to the energy changes occurring in the Carnot s cycle. 

Theorem.—A process yields the maximum amount of available energy when it is conducted reversibly.—Proof. If the change is isothermal, this is a consequence of Moutier s theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectlyJ reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot s cycles (Fig 11). 

In a simple Carnot s cycle, in which heat QA is absorbed from the source at temperature TA, and heat QB is emitted to the refrigerator at temperature TB, we have ... 

In the operations constituting a Carnot s cycle, changes of Q and T occur separately. In the majority of cases both these changes occur together, so that the temperature of the working substance may be regarded as a function of the time. Equation (4) therefore requires extension, and this was effected by Lord Kelvin in May, 1854, in the following way ... 

As a particular case we may instance the Carnot s cycle ABCD, Fig. 8. c... 

The area is positive if traced out clockwise. Since the heat absorbed in the cycle is equal to the work done, the areas of the Carnot s cycle on the (p, v) and (S, T) diagrams are equal. This may be generalised to apply to any reversible cycle where the only external work is done by expansion. 

This follows from the result established for the Carnot s cycle ... 

In a Carnot s cycle, the entropy Qi/Ti is taken from the hot reservoir, and the entropy Q2/T2 is given up to the cold reservoir, and no other entropy change occurs anywhere else. Since these two quantities of entropy are equal and opposite, the entropy. change in the hot reservoir is exactly balanced, or, to use an expression of Clausius, is compensated by an equivalent change in the cold reservoir. Again, in any reversible cycle there is on the whole no production of entropy so that all the changes are compensated. 

Example.—An ideal gas of constant specific heat is taken round a reversible Carnot s cycle, represented by four curves (Fig. 22) ... 

A mixture of solid and liquid in equilibrium at a temperature (T — 8T) under a pressure (p — bp), is taken round a small Carnot s cycle ABCD. 

If the changes of volume are executed very rapidly, they may be made adiabatic, and a Carnot s cycle may be performed with the apparatus. We take Y, the total volume of the system in the cylinder, as the abscissa, and P, the osmotic pressure, as ordinate. Let... 

Carnot s cycle of Fig. 4.3 is certainly a remarkably simple realization of a heat engine. Can any other engine do better Surprisingly, Carnot concluded that the answer must be No Carnot s general conclusion can be summarized in the following statement ... 

Carnot s cycle is the most efficient possible heat engine. 

He was a practical electrician but fond of whiskey, a heavy, red-haired brute with irregular teeth. He doubted the existence of the Deity but accepted Carnot s cycle, and he had read Shakespeare and found him weak in chemistry. 

Since all reversible heat engines working between the same two temperatures will have the same efficiencies, we can conclude that their efficiencies depend only upon the two temperatures between which they work. For further thermodynamic consideration it is, therefore, sufficient that we consider that type of reversible machine, which will lend itself to simple thermodynamic treatment. A machine employing Carnot s cycle is of such a type. 

Now the system has reached the original state and the area ABCD represents the net work done by the system, which is not zero. Thus, the system is a heat engine, which has worked in a cycle in a reversible manner and has converted some heat into work. The cycle the system has undergone is called the Carnot s Cycle . 

In the discussion in the last section on Carnot s Cycle, we have noted that for a full cycle of operation of a reversible heat engine... 

Figure 6.1 Any cyclic reversible heat engine can be seen as comprising a large number of Carnot s Cycles...

As an example we would try to conceive of a Carnot s cycle without using ideal gas as the system. Let the container in Fig. 4.1 contain water and steam in equilibrium instead of an ideal gas. Here pressure would be 1 atmosphere if temperature was 100 °C. If heat is supplied to the system, more of water would get converted to steam. It is easy to conceive that, if heat is transferred infinitesimally slowly, heat transfer can be carried out in thermodynamically reversible manner with corresponding increase in volume. By releasing pressure, also in a reversible manner, more of water gets vaporised and volume increases further to point C (Fig. 6.6) at reduced pressure and temperature. Thereafter the system can be made to lose heat reversibly at the lower temperature, which would make some steam to condense to liquid water with reduction in volume. 

Figure 6.6 Carnot s Cycle with water vapour system...

Fig. 17 is a graphical representation of Carnot s cycle. The isothermals are horizontal lines, and the adiabatics are curves... 

We thus obtain the strikingly simple result that the efficiency of Carnot s cycle/ and hence, for the reasons above stated, the maximum efficiency of every other periodic machine, is directly proportional to the difference between the temperatures of the source and sink, and inversely proportional to the temperature of the source. 

Carnot s cycle has been thus far discussed for an ideal gas. Similar reversible cycles can be performed on other materials, including solids and liquids, and the efficiency of these cycles determined. The importance of the reversible cycle for the ideal gas is that, as has just been seen, it gives us an extremely simple expression for the efficiency, namely (T, Tc)/Th. Similarly, a theorem of Carnot shows that the efficiency of all reversible cyclj S,.Qperating.betwe.en, t % is the same, namely... 

---